Log-concave Probability Measures Research Articles

Half-space depth of log-concave probability measures

Given a probability measure mu on {{mathbb {R}}}^n, Tukey’s half-space depth is defined for any xin {{mathbb {R}}}^n by varphi _{mu }(x)=inf {mu (H):Hin {{{mathcal {H}}}}(x)}, where mathcal{H}(x) is the set of all half-spaces H of {{mathbb {R}}}^n containing x. We show that if mu is a non-degenerate log-concave probability measure on {{mathbb {R}}}^n then e-c1n⩽∫Rnφμ(x)dμ(x)⩽e-c2n/Lμ2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} e^{-c_1n}\\leqslant \\int _{{\\mathbb {R}}^n}\\varphi _{\\mu }(x)\\,d\\mu (x) \\leqslant e^{-c_2n/L_{\\mu }^2} \\end{aligned}$$\\end{document}where L_{mu } is the isotropic constant of mu and c_1,c_2>0 are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of L_q-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.

Threshold for the expected measure of random polytopes

Let μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} be a log-concave probability measure on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^n$$\\end{document} and for any N>n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N>n$$\\end{document} consider the random polytope KN=conv{X1,…,XN}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N=\ extrm{conv}\\{X_1,\\ldots ,X_N\\}$$\\end{document}, where X1,X2,…\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_1,X_2,\\ldots $$\\end{document} are independent random points in Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb R}^n$$\\end{document} distributed according to μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. We study the question if there exists a threshold for the expected measure of KN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N$$\\end{document}. Our approach is based on the Cramer transform Λμ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _{\\mu }^{*}$$\\end{document} of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}. We examine the existence of moments of all orders for Λμ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda _{\\mu }^{*}$$\\end{document} and establish, under some conditions, a sharp threshold for the expectation EμN[μ(KN)]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {E}}_{\\mu ^N}[\\mu (K_N)]$$\\end{document} of the measure of KN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K_N$$\\end{document}: it is close to 0 if lnN≪Eμ(Λμ∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln N\\ll {\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*})$$\\end{document} and close to 1 if lnN≫Eμ(Λμ∗)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ln N\\gg {\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*})$$\\end{document}. The main condition is that the parameter β(μ)=Varμ(Λμ∗)/(Eμ(Λμ∗))2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta (\\mu )=\ extrm{Var}_{\\mu }(\\Lambda _{\\mu }^{*})/({\\mathbb {E}}_{\\mu }(\\Lambda _{\\mu }^{*}))^2$$\\end{document} should be small.

SIGEST

The SIGEST article in this issue is “A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau,” by Alain Durmus, Éric Moulines, and Marcelo Pereyra. The authors provide new algorithms to sample from high-dimensional log-concave probability measures, where they combine Moreau--Yosida envelopes with the Euler--Maruyama discretization of Langevin diffusions. This allows for an efficient Markov chain Monte Carlo methodology that is applicable to inverse problems arising in imaging sciences. Asymptotic and nonasymptotic convergence results are provided, along with extensive computational experiments on realistic imaging problems involving deconvolution and tomographic reconstruction. The original article, which appeared in the SIAM Journal on Imaging Sciences in 2018, has attracted substantial interest. In preparing this highlighted SIGEST version, the authors have expanded the introduction to make it accessible to a wide audience. The final section also discusses follow-up work arising from the original publication.

A proof of the Caffarelli contraction theorem via entropic regularization

We give a new proof of the Caffarelli contraction theorem, which states that the Brenier optimal transport map sending the standard Gaussian measure onto a uniformly log-concave probability measure is Lipschitz. The proof combines a recent variational characterization of Lipschitz transport map by the second author and Juillet with a convexity property of optimizers in the dual formulation of the entropy-regularized optimal transport (or Schrodinger) problem.

Normal approximation for weighted sums under a second-order correlation condition

Under correlation-type conditions, we derive an upper bound of order $(\log n)/n$ for the average Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law. The result is based on improved concentration inequalities on high-dimensional Euclidean spheres. Applications are illustrated on the example of log-concave probability measures.

Generalized dual Sudakov minoration via dimension-reduction—a program

We propose a program for establishing a conjectural extension to the class of (origin-symmetric) log-concave probability measures $\mu$, of the classical dual Sudakov Minoration on the expectation of the supremum of a Gaussian process: \begin{equation} \label{eq:abstract} M(Z_p(\mu), C \int ||x||_K d\mu \cdot K) \leq \exp(C p) \;\;\, \forall p \geq 1 . \end{equation} Here $K$ is an origin-symmetric convex body, $Z_p(\mu)$ is the $L_p$-centroid body associated to $\mu$, $M(A,B)$ is the packing-number of $B$ in $A$, and $C > 0$ is a universal constant. The Program consists of first establishing a Weak Generalized Dual Sudakov Minoration, involving the dimension $n$ of the ambient space, which is then self-improved to a dimension-free estimate after applying a dimension-reduction step. The latter step may be thought of as a conjectural small-ball one-sided variant of the Johnson--Lindenstrauss dimension-reduction lemma. We establish the Weak Generalized Dual Sudakov Minoration for a variety of log-concave probability measures and convex bodies (for instance, this step is fully resolved assuming a positive answer to the Slicing Problem). The Separation Dimension-Reduction step is fully established for ellipsoids and, up to logarithmic factors in the dimension, for cubes, resulting in a corresponding Generalized (regular) Dual Sudakov Minoration estimate for these bodies and arbitrary log-concave measures, which are shown to be (essentially) best-possible. Along the way, we establish a regular version of (\ref{eq:abstract}) for all $p \geq n$ and provide a new direct proof of Sudakov Minoration via The Program.

Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications

AbstractWe review some simple techniques based on monotonemass transport that allow us to obtain transport-type inequalities for any log-concave probability measures, and formore generalmeasures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.

Remarks on curvature in the transportation metric

According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kahler–Einstein equation e Φ = detD 2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D 2Φ, e Φ dx) has a non-positive Bakry–Emery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Emery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.

Transport inequalities for log-concave measures, quantitative forms and applications

We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.

Maximal surface area of polytopes with respect to log-concave rotation invariant measures

It was shown in [19] that the maximal surface area of a convex set in Rn with respect to a rotation invariant log-concave probability measure γ is of order nVar|X|4E|X|, where X is a random vector in Rn distributed with respect to γ. In the present paper we discuss surface area of convex polytopes PK with K facets. We find tight bounds on the maximal surface area of PK in terms of K. We show that γ(∂PK)≲nE|X|⋅log⁡K⋅log⁡n for all K. This bound is better than the general bound for all K∈[2,ecVar|X|]. Moreover, for all K in that range the bound is exact up to a factor of log⁡n: for each K∈[2,ecVar|X|] there exists a polytope PK with at most K facets such that γ(∂PK)≳nE|X|log⁡K.

Eigenvalue distribution of optimal transportation

We investigate the Brenier map $\nabla \Phi$ between the uniform measures on two convex domains in $\mathbb{R}^n$ or more generally, between two log-concave probability measures on $\mathbb{R}^n$. We show that the eigenvalues of the Hessian matrix $D^2 \Phi$ exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension $n$.

KLS-type isoperimetric bounds for log-concave probability measures

The paper considers geometric lower bounds on the isoperimetric constant for logarithmically concave probability measures, extending and refining some results by Kannan et al. (Discret Comput Geom 13:541–559, 1995).

Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension

We consider an elliptic Kolmogorov equationu Ku = f in a separable Hilbert space H. The Kolmogorov operator K is associated to an infinite dimensional convex gradient system: dX = (AX DU(X))dt + dW(t), where A is a self-adjoint operator in H and U is a convex lower semicontinuous function. Under mild assumptions we prove that for � > 0 and f 2 L 2 (H,�) the weak solution u belongs to the Sobolev space W 2,2 (H,�), whereis the log-concave probability measure of the system. Moreover maximal estimates on the gradient of u are proved. The maximal regularity results are used in the study of perturbed non gradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction-diffusion and Cahn-Hilliard stochastic PDEs.

Neighborhoods on the Grassmannian of marginals with bounded isotropic constant

We show that for any isotropic log-concave probability measure μ on Rn, for every ε>0, every 1⩽k⩽n and any E∈Gn,k there exists F∈Gn,k with d(E,F)<ε and LπFμ<C/ε.

Spectral gap for spherically symmetric log-concave probability measures, and beyond

Let μ be a probability measure on R n ( n ≥ 2 ) with Lebesgue density proportional to e − V ( ‖ x ‖ ) , where V : R + → R is a smooth convex potential. We show that the associated spectral gap in L 2 ( μ ) lies between ( n − 1 ) / ∫ R n ‖ x ‖ 2 μ ( d x ) and n / ∫ R n ‖ x ‖ 2 μ ( d x ) , improving a well-known two-sided estimate due to Bobkov. Our Markovian approach is remarkably simple and is sufficiently robust to be extended beyond the log-concave case, at the price of potentially modifying the underlying operator in the energy, leading to weighted Poincaré inequalities. All our results are illustrated by some classical and less classical examples.

Error bounds for Metropolis–Hastings algorithms applied to perturbations of Gaussian measures in high dimensions

The Metropolis-adjusted Langevin algorithm (MALA) is a Metropolis-Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich-Rubinstein-Wasserstein distance of the MALA chain with semi-implicit Euler proposals applied to log-concave probability measures that have a density w.r.t. a Gaussian reference measure. For sufficiently "regular" densities, the estimates are dimension-independent, and they hold for sufficiently small step sizes $h$ that do not depend on the dimension either. In the limit $h\downarrow0$, the bounds approach the known optimal contraction rates for overdamped Langevin diffusions in a convex potential. A similar approach also applies to Metropolis-Hastings chains with Ornstein-Uhlenbeck proposals. In this case, the resulting estimates are still independent of the dimension but less optimal, reflecting the fact that MALA is a higher order approximation of the diffusion limit than Metropolis-Hastings with Ornstein-Uhlenbeck proposals.

The first variation of the total mass of log-concave functions and related inequalities

On the class of log-concave functions on Rn, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such a first variation, which suggest to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski’s first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski’s problem for convex bodies, and prove some partial results towards its solution.

$𝜓_{𝛼}$-estimates for marginals of log-concave probability measures

We show that a random marginal πF (μ) of an isotropic log-concave probability measure μ on R exhibits better ψα-behavior. For a natural variant ψ′ α of the standard ψα-norm we show the following: (i) If k ≤ √ n, then for a random F ∈ Gn,k we have that πF (μ) is a ψ′ 2measure. We complement this result by showing that a random πF (μ) is, at the same time, supergaussian. (ii) If k = n, 1 2 < δ < 1, then for a random F ∈ Gn,k we have that πF (μ) is a ψ′ α(δ)-measure, where α(δ) = 2δ 3δ−1 .

Small ball probability estimates for log-concave measures

We establish a small ball probability inequality for isotropic log \log -concave probability measures: there exist absolute constants c 1 , c 2 &gt; 0 c_{1}, c_{2}&gt;0 such that if X X is an isotropic log \log -concave random vector in R n {\mathbb R}^n with ψ 2 \psi _{2} constant and bounded by b b and if A A is a non-zero n × n n\times n matrix, then for every ε ∈ ( 0 , c 1 ) \varepsilon \in (0,c_{1}) and y ∈ R n y \in \mathbb R^n , \[ P ( ‖ A x − y ‖ 2 ≤ ε ‖ A ‖ HS ) ≤ ε ( c 2 b ‖ A ‖ HS ‖ A ‖ op ) 2 , \mathbb P \left ( \|Ax-y\|_{2} \leq \varepsilon \|A\|_{\textrm {HS}} \right ) \leq \varepsilon ^{\big ( \frac {c_{2}}{b}\frac {\|A\|_{\textrm {HS}}}{\|A\|_{\textrm {op}}} \big )^{2} }, \] where c 1 , c 2 &gt; 0 c_{1}, c_{2}&gt;0 are absolute constants.

Isoperimetry for spherically symmetric log-concave probability measures

We prove an isoperimetric inequality for probability measures $\\mu$ on $\\mathbb{R}^n$ with density proportional to $\\exp(-\\phi(\\lambda |x|))$, where $|x|$ is the euclidean norm on $\\mathbb{R}^n$ and $\\phi$ is a non-decreasing convex function. It applies in particular when $\\phi(x)=x^\\alpha$ with $\\alpha \\ge 1$. Under mild assumptions on $\\phi$, the inequality is dimension-free if $\\lambda$ is chosen such that the covariance of $\\mu$ is the identity.